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Book Review*: Category Theory in Context *by Emily Riehl

When I was an undergraduate, I helped organize a mathematical society at Queens College of the City University of New York. It had been decades since there was such a society at my alma mater, which hasn’t exactly been known for producing world-class scientists and mathematicians to begin with. It’s produced a few, in all fairness-but it ain’t exactly MIT and never has been. Most of the department thought it was amusing, but me and a few other students didn’t care.

A year or so went by where we barely kept it afloat until CUNY began the Honors College, which actively began recruiting among the best students from local high schools. After that, the organization took off as some immensely talented students began showing up in our lower division classes. Simultaneously, some young faculty arrived to replace the retiring old hats, such as John Terrilla and Christopher Hanusa. For a few years, we had a small but vibrant group of brilliant undergraduates who made it hopeful this little department could become an oasis of learning and discovery in the academic desert of the constantly strapped CUNY that could actually produce mathematicians and scientists of note.

(Of course, CUNY being CUNY, once those students moved on to graduate school, the CUNY Powers- That-Be in Albany and the Governor’s office couldn’t wait to mock the Math Club and slash funding to the department that benefited so that we didn’t dare have the arrogance or resources to imagine ourselves in the same league as Harvard, Yale or even NYU or Berkeley. Breaks my heart, but not surprising. )

During those halcyon few years, the department tried it’s best to support those students by running some actual challenging courses beyond linear algebra and calculus.

It was right about then that I ran head first into the Promethean Galaxy Wall called category theory for the first time.

I first met categories in one of those courses, John Terrilla’s Math 703 course-a serious first course in topology. John was a PhD student of John Schlessinger and his areas of research-then as now-included deformation theory and topology. So he was definitely a missionary of the Cult of The Diagram Chase. With brilliant and impressionable young students like Joseph Hirsh, Josiah Sugarman, Franklin Lee and Arthur Parzygnat in his courses, he had many willing natives to indoctrinate.

Sadly, the author was a native who couldn’t let The Old Ways go.

Category theory is largely an abstract generalization of the entire structure of modern mathematics where functions rather than sets and their elements are the atomic components from which all else is constructed. While I understood the basic definitions and ideas of categories, morphisms, functors, duals and adjoints, being originally trained in rigorous mathematics via naïve set theory, it was very tough for me transition to-as the late, great Joseph Rotman wonderfully described it-“living without elements”. It was very hard to let go of the ideas of mathematics as being woven from collections of various “sizes” and “structures” that were composed of “elements”, functions were a special kind of Cartesian products of such sets,etc. The central idea of category theory is to reverse the logic of this view of mathematics and take functions-more specifically, arrows or morphisms-as the irreducible components from which everything else is synthesized.

To say it was frustrating for me would be an understatement-especially since most practitioners of categorical arguments consider them obvious to anyone with a functioning brain. Making it even more self-doubting of one’s skills was the fact that not only did most of my classmates have no trouble understanding it, they seemed to favor it by far over set theoretic approaches.

I continued to struggle with categories and its bizarre mysticism of commuting triangles and squares until well into my graduate training. Part of this struggle resulted from the fact there wasn’t really a lot of beginners’ texts on category theory from which a baffled beginner could seek guidance. My friend Joey Hirsh was one of Terrilla’s early converts to this strange religion. He was then a brilliant sophomore taking courses with me-first in Kenneth Kramer’s honors algebra course and later in virtually all of Terilla’s advanced courses. (After completing his PhD under John’s guidance, he’s now an equally brilliant post-doc at MIT specializing in algebraic topology.) He tried his best to help me understand the magic, but I was blind to the light of it. Joey saw commutative diagrams as directed graphs of a particular kind and that’s how he tried to teach it to me. Since I was pretty fluent in graph theory, that approach helped a little-but ultimately not much.

What

**did**help me understand-in addition to working through several sessions in John’s office-was looking at as many examples of categories, morphisms and functors as I could find. The example of groupoids in particular was helpful to me because it was a simple generalization of the definition of a group. A groupoid is simply a category where all the morphisms are invertible i.e. are isomorphisms. It was really while studying the groupoid and its’ role in topology using Ronald Brown’s wonderful book that the proverbial lightbulb went off over my head: Category theory strips all but the essential properties that link all the objects in a single category and their morphisms.

While a set-theoretic foundation for mathematical constructions certainly makes them precise and rigorous, it also frequently obscures what their most significant properties are in the complexity of the machinery. A simple example is that of an isomorphism. The set theoretic definition: A function f:A → B i.e. a nonempty subset of the set of ordered pairs A × B such that no 2 different ordered pairs have the same first member,which is injective and surjective i.e. for every x,y ɛ A, f(x) = f(y) and for every b in B, there exists x in A such that b= f(x). A lot to unpack there, even though it’s fairly straightforward. Using this definition, we can prove if f is an isomorphism, there exists a unique g: B→A such that g o f (x) = x for every x in A. We call g the inverse of f. We can then prove f is an isomorphism iff f has an inverse function i.e. it’s invertible. Comparing with the categorical definition of an isomorphism: f: A →B is an isomorphism of the categories A and B if f is invertible, i.e. there exists a morphism g: B→A such that f o g = I ( B) and g o f = I (A) where I is the identity morphism for each category. While working with arrows doesn’t always lead to such simplifications, since most of the important properties of mathematical objects either depend on the functions between objects rather than the structure of the objects themselves or can be “coded” by functions, categorical language does result in simplifications a great deal of the time.

Since then, category theory has continued to vex me even though I think I’ve mastered the basic ideas of it. What I’ve hoped for along the way-not only for myself but future students-is a textbook that would allow a strong mathematics undergraduate or first year graduate student to master this odd but so important subject.

The classic mathematical introduction to the subject is, of course, Saunders McLane’s

*Categories For the Working Mathematician*, by one of the subject’s founding fathers. But the title of that book is certainly to be taken literally-MacLane is pitched at advanced graduate students and PhDs whose original training had not included categories. I remember trying to read it as an undergraduate and after 60 pages, I was actually exhausted. Literally mentally exhausted. It’s incredibly dense, packing an enormous amount into each paragraph while providing very little motivation and few examples. While McLane was certainly a master and writes very well, this is not a friendly text on a subject that isn’t easy to begin with. The title is as much a warning to the prospective reader as a description.

It also should be noted there have been a number of advances in the subject since the latest edition of MacLane, such as higher category theory, abstract homotopy theory and higher topos theory. Much more significantly, a huge number of important applications of categories and functors have arisen in a number of fields, from computer science to linguistics to neuroscience. So MacLane no longer represents the frontiers of the subject. There are more far accessible introductions now-some of which I’ll discuss in a future post-but most of these other introductions are not for strong mathematics students. They are introductions for graduate students in other fields like philosophy, linguistics and computer science who need an understanding of category theory to pursue research. The more mathematically oriented of such books could also be used as undergraduate mathematics student texts.

What’s needed for the audience I have in mind is a successor textbook to MacLane which a) covers all the major definitions and theorems of categories and their morphisms with good exercises b) provides a large stock of examples from various fields of mathematics to both motivate and clarify these ideas and c) does (a) and (b) in a comprehensive, well organized and well-written way but isn’t too long winded and (d) assumes a reasonable level of prerequisite knowledge from the students. In other words, we need a text which is pitched at an intermediate level between the mathematics undergraduate/non-mathematics graduate student books and Mac Lane.

Which brings us to this little green volume from Dover Books’ new Aurora line of original textbooks-

*Category Theory In Context*by Emily Riehl.

Riehl is a young mathematician at John Hopkins who’s been developing the lecture notes from which this book is based from courses in category theory she’s taught at both Harvard and John Hopkins for strong undergraduates and first year graduate students since 2015. I’ve seen the early draft versions of the notes since they began at her homepage and I was incredibly excited by what I saw gestating there. When the book was finished and published with surprising speed in an inexpensive Dover paperback-which made me even more excited, since it meant the book would be affordable for most students-I didn’t hesitate to buy it and begin reading it voraciously.

So what’s the verdict, you ask? Is it a good book someone could use to either learn or teach category theory?

Absolutely not. It’s

**not**a good book.

__It’s a game changing textbook.__Riehl has written a textbook that will not only become a classic in short order, but one that will change the teaching of category theory in universities across the world at every level. It’s richly written, reasonably detailed, crystal clear, completely up to date and wonderfully organized. It’s going to encourage many more math departments to begin to offer category theory to strong undergraduates and graduate students in both regular courses and independent reading seminars. It’s going to make teachers of graduate courses at relatively weak programs much more comfortable using diagram chasing in their presentation, it’s going to land on the required reading lists of the top graduate programs like Harvard and Columbia for the suggested background of applying undergraduates.

In short, it’s going to raise one hell of a noise once people become familiar with it.

I firmly believe Riehl’s book is going to replace MacLane’s as the definitive textbook on the subject for advanced pure mathematics students and it will do so relatively quickly.

__Yes, it’s__

**that**good.And now an overview of this future classic.

The book begins with a lengthy Preface, which is actually a wonderful short essay on category theory itself, providing a preview of much of the books content as well as stating important theorems. It’s in this preface that Riehl sets the tone for what follows with her beautiful writing style and wonderfully intuitive priming that sets the table for later rigorous definitions and proofs.

Chapter 1 sets the basic definitions and theorems of category theory: categories, morphisms, functors, natural transformations, abstract vs. concrete categories, duality and opposite categories, covariance and contravariance of functors and natural transformations. This chapter sets the tone for the rest of the book-there may be more examples of basic structures in category theory in this first chapter alone then in all other books on the subject published to date combined. More on that later. It also includes a rigorous discussion of diagrams and a glimpse of higher category theory through 2-categories.

Chapter 2 discusses universal properties of categories as encoded in natural transformations. The ultimate goal of this chapter is to state, prove and illustrate most of the important consequences of Yoneda’s Lemma, which is essentially “The Fundamental Theorem of Category Theory”. Yoneda’s lemma says that if certain conditions are met on a category C and there is a functor F: C→Set where a ɛ C, there exists an isomorphism functor from the category of all natural transformations on F to the image category F (A). This is really a crude statement of the result, which is actually a good deal more sophisticated than this. To state it precisely requires the definition of a functor being represented by an object in C, which in turn requires initial and terminal objects in a commutative diagram. These and all the associated machinery is defined and developed here.

Chapter 3 discusses a powerful generalization of topological spaces and their subspaces: limits and colimits in categories. This is where the precise formulation of a commutative diagram really comes into it's own here. A lot of other books shy away from this critical spade work and it many other sources, it really makes limits confusing. Riehl makes you understand in this chapter how difficult it is to develop and understand categorical limits rigorously without it.

Chapter 4 defines and explains the rather subtle but quite important concept of adjunction. More sophisticated concepts of modern category theory hinge heavily on adjunction, so Riehl has her work cut out for her in making this chapter comprehensible. She has a very insightful way of motivating this concept: it can be thought of as the inverse of the forgetful functor’s action on a small category into the category of sets. As the forgetful functor strips all structure away from a defined category to produce a set with no structure, an adjunction takes a “free” construction of sets and builds a specific category with its’ expected structure. This is not an easy concept to for a beginner to grasp. This is where the plethora of examples the author provides in concert with a precise formulation is extremely clarifying.

The rest of the book tackles advanced concepts that a graduate student needs for forays into the current frontiers of modern algebra and topology as well as higher category theory. Chapter 5 discusses monads and their algebras. Chapter 6 discusses Kan extensions and their role as a unifying concept in category theory, where most of the concepts of the previous 5 chapters can be expressed as Kan extensions.

Before I go any deeper into this review, I don’t want to give the impression this is bathroom reading or something you can zip through on the bus on the way to class. It’s anything but. It’s certainly easier and much more accessible then MacLane. But that’s a little like saying the annotated Complete William Shakespeare is much more accessible than the original 15th century English versions. It’s true, but that’s hardly saying it’s an easy read. However, this is category theory. Like its’ brethren subjects in the foundations of mathematics, mathematical logic and axiomatic set theory, unless one gives a very shallow and cursory treatment, there’s really no way to make the subject easy to digest. What a reader can and should expect from a well-written treatment of such a subject is that his or her effort and focus will be rewarded with a deep and thorough understanding of the material that will allow them to study most advanced treatises and research papers on the subject without significant effort.

Riehl absolutely delivers on that here.

If a student is making the effort to seriously read an advanced mathematics textbook, the 2 most important qualities it must have are readability and clarity. It’s especially true if they’re brave enough to try and do so independently without a teacher. This book is brimming with both, which is quite a feat in a book at this level. Riehl’s prose is wonderful and lively, as well as possessing tremendous professional depth. It’s also concise in the best possible sense of the word-there’s virtually no irrelevant expository fat in the book. Everything in the book is important to the presentation, everything opens up a new perspective-however minor-on the material under consideration.

But what’s most striking about the book- indeed, its single most unique attribute and what raises it head and shoulders above most other textbooks written at this level- is the examples. Most practitioners of mathematics fall into 2 camps on explicit, specific examples in a mathematics textbook: One camp is the “Bourbakian” camp-which believes all mathematics, regardless of audience, is done at the highest level of generality and abstraction and that all concrete examples can and should automatically “drop out” as special cases of the powerful machinery. The other camp believes in the converse approach: We should begin with as many specific, concrete examples as possible and then prove general statements as a “big picture” statement encapsulating them all as special cases. I’m firmly in the second camp. Although everyone has different innate processing for mathematics, one doesn’t have to be a cognitive scientist to see humans tend to learn by going from the specific to the general. While the most abstract presentation may benefit experienced mathematicians, I believe for all but the most gifted students, the latter approach would be most beneficial.

I also think there’s considerable evidence even the most hardened abstractionists in mathematics understand this. The pre-eminent Bourbaki Jean Dieudonne, in his classic textbook, Infinitesimal Calculus, put the more difficult proofs in small print for the beginning student to avoid on first pass. Also, towards the end of his remarkable career as a textbook author, Serge Lang rewrote his classic textbook on differentiable manifolds from an infinite dimensional Banach space setting to “ordinary” finite-dimensional vector spaces to make the latest version of the book accessible to a much larger audience of beginners. With most mathematics, there’s a wide spectrum of approaches one can take between the 2 above extremes. For example, one approach instead of dealing with specific examples, you can deliberately state and prove theorems in selected generality i.e. instead of proving the Heine-Borel theorem on complete and totally bounded metric spaces, we can prove it in the case of closed and bounded subsets of Euclidean spaces.

When one is dealing with category theory, one is in an interesting predicament. The entire point of the subject is to present mathematics as abstractly as possible and remove all nonessential properties. As a result, any “middle ground” approach isn’t really an option. You basically can either focus on concrete examples of categories and morphisms and build the general structures around them or state and prove everything in full generality very tersely with no or few examples ,with many major results shunted to the exercises. The latter approach is what MacLane does-which is what makes it so arid and forbidding. But given the fact MacLane was writing for strong or advanced graduate students or PhDs, this isn’t really surprising.

Riehl takes the former approach with gusto. It is brazenly, unapologetically and passionately example-driven. And it does so without sacrificing the least bit of rigor or abstraction. This is the “context” referred to in the title. All definitions, theorems and proofs are deeply embedded in a framework of examples.

Many, many, many examples.

With a very few possible exceptions such as E.B. Vinberg’s

*A Course In Algebra*, John and Barbara Hubbard’s

*Vector Calculus,Linear Algebra And Differential Forms*and a handful of others, I’ve never seen any textbook with as many beautifully detailed and presented examples as Riehl’s book.

**Never.**

You’re actually stunned by not only how many examples there are in the book, but how diverse and insightful many of them are. These examples are drawn from virtually all mathematical disciplines and vary enormously in level of difficulty. And how original a number of them are. I certainly haven’t thought of many of them in the full context she presents and I’m willing to bet even professional mathematicians will be seeing many of them for the first time.

She also gives credit “where credit is due” for those who assisted or inspired both this truly incredible number and diversity of examples she’s collected and presented as well as the nearly equinumerous exercises in the text. She credits quite literally dozens of mathematicians she’s communicated with while writing it and past students in her classes. This includes, surprisingly, the aforementioned John Terilla. The mathematical academic world is a small category, indeed!

In fact, I had a lot of trouble picking my favorites to quote from. So I decided to make it easy for myself. I just picked my favorites from the set of examples she presents in Chapter 1. Keep in mind these are just from the first chapter and the density of examples in the book is uniform throughout.

*Example 1.1.3. Many familiar varieties of mathematical objects assemble into a category.*

*i) Set has sets as its objects and functions, with specified domain and codomain, as*

*its morphisms.*

*(ii) Top has topological spaces as its objects and continuous functions as its morphisms.*

*(iii) Set and Top have sets or spaces with a specified basepoint as objects and basepoint preserving (continuous) functions as morphisms.*

*(iv) Group has groups as objects and group homomorphisms as morphisms. This example lent the general term “morphisms” to the data of an abstract category. The categories Ring of associative and unital rings and ring homomorphisms and Field of fields and field homomorphisms are defined similarly.*

*(v) For a fixed unital but not necessarily commutative ring R, Mod R is the category of*

*left R-modules and R-module homomorphisms. This category is denoted by Vect k*

*when the ring happens to be a field k and abbreviated as Ab in the case of Mod Z, as*

*a Z-module is precisely an abelian group.*

*(vi) Graph has graphs as objects and graph morphisms (functions carrying vertices to*

*vertices and edges to edges, preserving incidence relations) as morphisms. In the*

*variant DirGraph, objects are directed graphs, whose edges are now depicted as*

*arrows, and morphisms are directed graph morphisms, which must preserve sources*

*and targets.*

*(vii) Man has smooth (i.e., infinitely differentiable) manifolds as objects and smooth maps*

*as morphisms.*

*(viii) Meas has measurable spaces as objects and measurable functions as morphisms.*

*(ix) Poset has partially-ordered sets as objects and order-preserving functions as morphisms.*

*(x) Ch R has chain complexes of R-modules as objects and chain homomorphisms as*

*morphisms.*

*(xi) For any signature σ specifying constant, function, and relation symbols, and for*

*any collection of well-formed sentences T in the first-order language associated to*

*σ, there is a category Model T whose objects are σ-structures that model T, i.e., sets*

*equipped with appropriate constants, relations, and functions satisfying the axioms*

*T. Morphisms are functions that preserve the specified constants, relations, and*

*functions, in the usual sense. Special cases include (iv), (v), (vi), (ix), and (x).*

*(iii) A poset (P; ≤_) (or, more generally, a preorder ) may be regarded as a category. The*

*elements of P are the objects of the category and there exists a unique morphism*

*x → y if and only if x ≤ y. Transitivity of the relation “≤” implies that the required*

*composite morphisms exist. Reflexivity implies that identity morphisms exist.*

*(iv) In particular, any ordinal α = { β|β <α} defines a category whose objects are the*

*smaller ordinals. For example, 0 is the category with no objects and no morphisms. 1*

*is the category with a single object and only its identity morphism. 2 is the category*

*with two objects and a single non-identity morphism, conventionally depicted as*

*0 → 1. is the category freely generated by the graph 0 → 1 →2 → 3 →…. in the sense that every non-identity morphism can be uniquely factored as a composite of morphisms in the displayed graph; a precise definition of the notion of free generation is given in Example 4.1.13.*

*The examples above are quoted from 2 pages in Riehl’s book.*

That’s right,

**2 pages.**

That’s how incredibly dense in examples the book is. While I was reading it, I couldn’t help wondering why no one had written such an example-driven account of category theory before. The subject is a framework for all mathematics. As a result, its’ threads are woven throughout the entire fabric of modern mathematics. So you’d think such an approach would be very-forgive the pun-natural. It may have to do with the fact that until very recently, this was considered a subject at the very apex of mathematical abstraction. As a result, its serious indoctrination was left to very advanced graduate students and research mathematicians. MacLane is clearly written for exactly this purpose. But as stated before, this is no longer an accurate picture of the subject because of the emergence of the many applications mentioned. In all honesty, even if this were not the case, one would think the realization of the interconnected web of categories and morphism woven throughout mathematics would have resulted in such an example-driven approach being written sooner. It’s a bit of a mystery to me why it’s taken so long for one to be written.

But Riehl has now finally written one-and at an advanced level, no less. As clichéd as it is, once again the immortal poem by Robert Frost’s last passage is most appropriate as postscript for Riehl’s success here:

*Two roads diverged in a wood, and I—*

*I took the one less traveled by,*

*And that has made all the difference.*

A few more of this remarkable treasure trove will drive the point home further-for these examples are far more diverse in nature then the above sampling would suggest. While there are many of the explicit “standard” kind one sees in undergraduate mathematics, there are also quite a few of more sophisticated bent.

Quite a few are examples that demonstrate how the categorical perspective shows some deep unexpected relations between familiar mathematical objects. One of my favorites is Example 1.4.9: The categorification of the natural numbers, which uses the concept of a discrete category introduced above. This extraordinary construction uses natural isomorphisms from various Cartesian products of sets in the category of finite sets and bijections, denoted Fin iso and a cardinality functor into a discrete category of the natural numbers ℕ. Then assigning natural numbers to cardinality of finite sets yields the usual rules of ordinary arithmetic. Reversing the functor direction gives the categorification of the natural numbers in Fin iso where the algebraic laws of arithmetic are derived from the more primitive isomorphisms on finite sets! I heartily recommend carefully going through this construction and mastering it. It’s harder than the usual inductive proofs, but well worth the effort. It connects arithmetic on ℕ to Cartesian product isomorphisms on finite sets in a very direct manner that would be far from obvious using set theoretic methods.

Another type of example Riehl likes to use are generalizations of famous standard theorems of mathematics in category theory. A surprising example which she proves in full:

*Proposition 1.5.12. Any connected groupoid is equivalent, as a category, to the automorphism*

*group of any of its objects.*

It should be fairly obvious to anyone with a knowledge of group theory that this is a categorical generalization of Cayley’s Isomorphism Theorem of Groups. She also shows how a categorical perspective provides very simple, general and concise proofs of a number of standard mathematical results, such as the following famous and important result:

*Theorem 1.3.3 (Brouwer Fixed Point Theorem). Any continuous endomorphism of a 2-*

*dimensional disk D2 has a fixed point.*

Another terrific and unexpected example given later is how the following fundamental result of linear algebra can be shown to be a trivial consequence of the Yoneda Lemma:

*Corollary 2.2.9. Every row operation on matrices with n rows is defined by left multiplication*

*by some n × n matrix, namely the matrix obtained by performing the row operation*

*on the identity matrix.*

Indeed, she proceeds to demonstrate that a number of famous results can be shown to be corollaries of Yoneda’s Lemma on the category in question-including the aforementioned Cayley’s theorem as a functor F: Group→ Set which is a composition of an automorphism from G into the right G-set and the forgetful functor onto Set.

A high powered proof indeed!

A few words on the exercises. While there’s quite a few of them, most are straightforward and not too difficult-they either derive further examples that are consequences of the ones given in the text or prove corollaries of the results proven in the chapter. They add more depth to both the exercises and the major results without being necessary to the flow of the text. While one does want to encourage students in mathematics courses-particularly advanced ones-to do as many exercises as possible, I don’t think making the exercises an essential part of the flow of the text is mandatory for learning as long. If the subject matter is particularly challenging, as is the case here, this will frustrate the beginner. We want to encourage students to push through difficult material. I’m pretty sure putting major substantial results for the beginner to prove by themselves does the opposite. We should reserve such exercises for experienced graduate students. However, on the opposite end of the Gauss bell curve, we shouldn’t make exercises so easy they’re useless as practice or learning tools either. Riehl does a very good job of pitching exercises that are just difficult enough to be instructive, but not so difficult the student will throw the book across the room after 7 hours on a single exercise. This makes the exercises just as informative as the examples and there are nearly as many.

As wonderful as the book is, I do have some minor quibbles with it. No textbook is perfect and this one is no different. (Not even the author of a text should consider it perfect. Which should be the real reason authors should write later editions, namely to create a finite sequence of editions that converges to perfection as n → + ∞. Sadly, it usually the last reason even considered by publishers.)

Firstly, I wish in a book this wonderfully researched, that Riehl had incorporated more historical notes. The book actually opens with a detailed account of the birth of category theory-which began as most new branches of mathematics begin, with an attempt to solve a specific problem. In category theory’s case, it grew from MacLane and Samuel Eilenberg’s attempt in 1941 to give a general formulation of the universal coefficient theorem of algebraic topology. This is a wonderfully detailed and researched vignette one rarely sees in textbooks and is to be heartily praised. Unfortunately, it’s a relative anomaly in the book. Another particularly nice example: Section 2.2 on the statement of the Yoneda lemma begins with a quote from a paper by MacLane on how he named the lemma during Yoneda’s visit to France. I wish she’d expanded considerably on this quote in the chapter (although she does include full references in a footnote). I wish she’d inserted many narrative sidebars like the one that opens the book, embedding this difficult material into a rich historical context. The history of scientific and mathematical thought is an amazing tapestry filled with amazing people and events lush with life. I always encourage teachers to mine it as thoroughly as possible. Not only does this bring the subject to life, it gives full answers to the many “why” questions students have. Sooner or later, some student is going to ask you, “Why do we care?” It’s really hard for them to dispute the significance of a subject when you make them walk the path of those who created it-when you show them what lead them to create these concepts. (Of course, there’s always going to be the apathetic sociopath student who’s just passing time until they get into Harvard law or medical school, but we all have to deal with those.) There are some narrative detours like this in Riehl-I just wish there’d been quite a few more.

My major quibble with the book is one I have with a lot of mathematics textbooks, particularly advanced ones. It’s particularly a problem with this one. In fact, I’ll state it as an axiom for future reference:

**The Mathemagician’s Axiom of Textbook Prerequisites: Let M be the minimum actual prerequisites for an average student being able to read and understand a given presentation of a subject in a mathematics textbook. Let A be the author’s stated prerequisites. Then usually:**

**A <<<<<<< M**Another possible reason, which happens quite a bit when the author has the good fortune to teach at an elite school, is they tend to vastly overestimate the skill of most beginning students who learn this material. Most students who will read the book are nowhere near as talented and/ or prepared as the students the author usually has in his or her classes. Also, mathematicians that teach at such schools were usually gifted students themselves. So even if they themselves are solid teachers-a skill which is not really encouraged at elite schools, sadly-they may think such minimal background is sufficient for all students. For example, it’s hard to imagine a student that doesn’t have a pretty strong background in linear algebra and basic geometry, as well as a reasonable experience writing proofs, being able to learn algebra from Micheal Artin’s

*Algebra*. Artin, however, has only calculus and some set theory as the prerequisites for the text. I’m sure he’s had many such gifted students in his algebra courses over the years at MIT who have been able to get by just fine with this background or less-but I seriously doubt the average mathematics undergraduate could.

Not only does Riehl teach at a fairly elite university, she was a gifted mathematics student in her day. She was an undergraduate at Harvard University, a visiting scholar at Cambridge University and got her PhD at The University of Chicago under Peter May. So one could certainly forgive her for overestimating the skill and preparation of her potential audience.

In this case, she states that nothing more is needed to learn from the book then basic set theory and logic. Which makes anyone reading the book fight not to laugh out loud. I can make this argument instantly by showing you the opening paragraph of the first page of Chapter 1:

*A group extension of an abelian group H by an abelian group G consists of a group E*

*together with an inclusion of G E as a normal subgroup and a surjective homomorphism*

*E _ H that displays H as the quotient group E/G. This data is typically displayed in a*

*diagram of group homomorphisms:*

*0 → G→ E → H → 0*

*A pair of group extensions E and E’ of G and H are considered to be equivalent whenever*

*there is an isomorphism E ≈ E’ that commutes with the inclusions of G and quotient maps*

*to H, in a sense that is made precise in §1.6. The set of equivalence classes of abelian*

*group extensions E of H by G defines an abelian group Ext(H,G).*

This is the first paragraph of the book.

Seriously.

Unless that introduction to set theory doubled as a serious introduction to group theory, the reader’s eyes are going to glaze over 4 words into reading it.

**The reader won’t even understand the words!**So Riehl’s claim of the demands the book makes on the reader are flat-out absurd.

In all fairness to Riehl-she herself pretty much admitted as much at Amazon to a reviewer there. Although she didn’t explicitly say so, I strongly suspect the “official” prerequisites printed in the finished book came from Dover and not her. Naturally they would minimize them because they don’t want to scare away any potential readers. This is why evaluating the level of academic textbooks needs to be left to academics and not corporate marketers.

Ok, throwing away the ridiculous claim, what do I think is the bare minimum background needed to be able to effectively read the book i.e. study and understand most of it? I think the student at absolute minimum would need:

1) A decent undergraduate course in abstract algebra, one that contains all the basic definitions and theorems of groups, rings and fields,

2) A good year-long undergraduate course in linear algebra up to and including diagonalization, the Cayley-Hamilton theorem and the Jordan form and

3) A good undergraduate course in topology that goes beyond basic point set topology to the fundamental group and basic homotopy in low dimensional spaces.

If one accepts premises (1)-(3), then this means the

**minimum**level the book could be used at by most mathematics majors would be early senior year. (Presumably honor students could use it earlier.) Such students could read the entire book with some effort, but doubtful they’ll be able to understand all the examples and do all the exercises. A first year graduate student could read and digest it completely as a supplement to standard first year courses. To me, the ideal student would be a senior mathematics major in his or her last undergrad year who’s completed the BS requirements, submitted his or her graduate school applications and Math Subject GRE scores and is trying to beef up the holes in their training while they await the verdict. Clearly, though, you’d like stronger young students with the correct background to pick it up.

But be that as it may, regardless of whose fault the understating of the prerequisites of the book is, it’s really important an honest assessment is made so that neither students or teachers misfire on its’ appropriate level. This is far too good a book to get a bad reputation for something this trivially basic. For future printings/editions, I strongly encourage Riehl to request a number of reviewers who are veteran mathematicians with extensive teaching experience and ask them what they think the appropriate background for the text is. I gave my own personal suggestions above.

But these are both minor quibbles and both are easily fixed. I believe Riehl has written the Great American Category Theory Book for the serious mathematics students at all levels. She is to be heartily commended on writing such a remarkable work and Dover is to be commended for making it available at such a reasonable price. Frankly, if this was the only original work published in Dover’s Aurora series, the line would be worth having. Here’s hoping in the future it produces many more affordable jewels like this one and Riehl's text-and hopefully many equally affordable revised editions-finds their way into the hands of many talented students both now and in the future as a result!

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